orbits. Each of these period n islands has asso
ciated with it a similar hierarchy of higherorder
island chains, and so on.
Consider an outermost intact invariant sur
face. A sufficient increase in the magnitude of
the perturbations will produce a tear in this sur
face. Under dynamical evolution of the sur
face, this tear will propagate and reproduce
over the surface, producing a fractal distribu
tion (Cantor set) of holes or gaps, known as a
cantorus (Percival, 1979; MacKay et al., 1984).
The surface is not completely destroyed, but
neither is it any longer impervious to diffusion
of orbits across it. There is a countable infinity
of holes. The stronger the perturbation(s), the
more "porous" is the cantorus. Thus, it is pos
sible for an orbit to be trapped in a region of
phase space, enclosed by a cantorus of small
porosity, for some time before encountering a
"hole" and escaping into a more unstable
region. Some orbits may escape quickly; oth
ers will be trapped for some time. Given an ini
tial population of such orbits, one would expect
a distribution of escape times perhaps Gaus
sian. The same argument can be used for orbits
near the secondary islands, which themselves
will be enclosed by cantori (and higherorder
islands, etc.).
Analytic calculation of diffusion rates
across hierarchies of cantori is very difficult.
As a trajectory passes through a cantorus, it
may be derailed to "shadow" a higherorder
island chain, which itself has a grid of cantori
gaps, and so on. Numerically, it can be shown
that the distribution of orbits initially in the
neighborhood of an island remaining in that
neighborhood (i.e., the "survival" probability) is
(2)
P
(
t
)
i t

b
for long times t (normalized by the orbit
period), where
1.4 for a small, isolated
island (Chirikov and Shepelyansky, 1984b;
Karney, 1983; Lichtenberg and Lieberman,
1992). For the main island of the standard
map (Chirikov, 1979),
1.45 (Chirikov and
Shepelyansky, 1984a; Murray, 1991). It
would be reasonable to wonder if the tori sur
rounding regular islands is the main barrier to
diffusion of a trajectory outward. However, it
appears that the sticking time to a given sur
face is the dominant process in impeding phase
space transport (Chirikov and Shepelyansky,
1984a,b; Murray, 1991). The departures of
diffusion from unimpeded random motion are
due to a small percentage of orbits that are
stuck around KAM surfaces bounding a region
of chaotic motion.
We may associate an escape past a particu
lar cantorus with an "event" in an asteroid
orbit, since (going back to the two degree of
freedom analogy) successive cantori are rapidly
more porous, leading quickly to the global sea
of chaos and therefore wild excursions of the
orbit. We propose that this is a reasonable pos
sibility for the mechanism displayed by the
outerbelt asteroids. The foregoing has been
wellestablished for simpler Hamiltonian sys
tems (cf. Lichtenberg and Lieberman, 1992;
Wiggins, 1990).
An alternative view is that KAM tori are
"sticky." Karney (1983) was the first to
numerically study the stickiness of KAM tori,
finding that longtime correlation functions for
a version of the standard mapping are strongly
governed by the dynamics near KAM surfaces.
Chaotic Motion in the Outer Asteroid Belt
page 9